On the Morse–Ekeland Index and Hamiltonian Oscillations

Abstract

In recent years, the Morse Index has been extensively used by many scientists. In order to study the convex Hamiltonian systems Ekeland used a Dual form of the least action principle, Morse theory and Thom’s transversely theorem, to show that on any prescribed energy level, either the closed trajectories are infinitely many, or they fulfill a resonance condition. It follows that the Morse–Ekeland index is an integer related to the linearized equation and it gives us information about closed trajectories (periodic trajectories). In this paper, we give some new properties of the Morse–Ekeland index. In particular, the Morse–Ekeland index of the perturbed Hamiltonian systems is defined and studied. After that, we show that the calculus of variations in mean, which is the natural generalisation of the calculi of variations for the almost periodic case, can’t allow us to define an index which generalized Ekeland’s for almost periodic trajectories.